Scientific Research

An Academic Publisher

The Role of Asymptotic Mean in the Geometric Theory of Asymptotic Expansions in the Real Domain

**Author(s)**Leave a comment

^{*}

We
call “asymptotic mean” (at +∞) of a real-valued function the number, supposed to exist, ,
and highlight its role in the geometric theory of asymptotic expansions in the
real domain of type (*) where the comparison functions ,
forming an asymptotic scale at +∞, belong to one of the three classes having a definite “type of variation” at +∞, slow, regular
or rapid. For regularly varying comparison functions we can characterize
the existence of an asymptotic expansion (*) by the nice property that a
certain quantity F（t) has an asymptotic mean at +∞. This quantity is
defined via a linear differential operator in *f* and admits of a remarkable geometric interpretation as it
measures the ordinate of the point wherein that special curve ,
which has a contact of order *n* - 1
with the graph of *f* at the generic
point *t*, intersects a fixed vertical
line, say *x* = *T*. Sufficient or necessary conditions hold true for the other two
classes. In this article we give results for two types of expansions already
studied in our current development of a general theory of asymptotic expansions
in the real domain, namely polynomial and two-term expansions.

KEYWORDS

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Advances in Pure Mathematics*,

**5**, 100-119. doi: 10.4236/apm.2015.52013.

[1] |
Granata, A. (2007) Polynomial Asymptotic Expansions in the Real Domain: The Geometric, the Factorizational, and the Stabilization Approaches. Analysis Mathematica, 33, 161-198. http://dx.doi.org/10.1007/s10476-007-0301-0 |

[2] |
Granata, A. (2010) The Problem of Differentiating an Asymptotic Expansion in Real Powers. Part I: Unsatisfactory or Partial Results by Classical Approaches. Analysis Mathematica, 36, 85-112. http://dx.doi.org/10.1007/s10476-010-0201-6 |

[3] |
Granata, A. (2010) The Problem of Differentiating an Asymptotic Expansion in Real Powers. Part II: Factorizational Theory. Analysis Mathematica, 36, 173-218. http://dx.doi.org/10.1007/s10476-010-0301-3 |

[4] |
Granata, A. (2011) Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part I: Two-Term Expansions of Differentiable Functions. Analysis Mathematica, 37, 245-287. (For an Enlarged Version with Corrected Misprints see: arxiv.org/abs/1405.6745v1 [mathCA]. http://dx.doi.org/10.1007/s10476-011-0402-7 |

[5] | Granata, A. (2014) Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II: The Factorizational Theory for Chebyshev Asymptotic Scales. Electronically Archived—arXiv: 1406.4321v2 [math.CA]. |

[6] |
Granata, A. (2015) The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results. Advances in Pure Mathematics, 5, 1-20. http://dx.doi.org/10.4236/apm.2015.51001 |

[7] | Haupt, O. (1922) über Asymptoten ebener Kurven. Journal für die Reine und Angewandte Mathematik, 152, 6-10; ibidem, 239. |

[8] | Sanders, J.A. and Verhulst, F. (1985) Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York. |

[9] | Corduneanu, C. (1968) Almost Periodic Functions. Interscience Publishers, New York. |

[10] |
Faedo, S. (1946) Il Teorema di Fuchs per le Equazioni Differenziali Lineari a Coefficienti non Analitici e Proprietà Asintotiche delle Soluzioni. Annali di Matematica Pura ed Applicata (the 4th Series), 25, 111-133. http://dx.doi.org/10.1007/BF02418080 |

[11] |
Hallam, T.G. (1967) Asymptotic Behavior of the Solutions of a Nonhomogeneous Singular Equation. Journal of Differential Equations, 3, 135-152. http://dx.doi.org/10.1016/0022-0396(67)90011-3 |

[12] | Hukuhara, M. (1934) Sur les Points Singuliers des équations Différentielles Linéaires; Domaine Réel. Journal of the Faculty of Science, Hokkaido University, Ser. I, 2, 13-88. |

[13] |
Ostrowski, A.M. (1951) Note on an Infinite Integral. Duke Mathematical Journal, 18, 355-359. http://dx.doi.org/10.1215/S0012-7094-51-01826-1 |

[14] |
Agnew, R.P. (1942) Limits of Integrals. Duke Mathematical Journal, 9, 10-19. http://dx.doi.org/10.1215/S0012-7094-42-00902-5 |

[15] | Hardy, G.H. (1911) Fourier’s Double Integral and the Theory of Divergent Integrals. Transactions of the Cambridge Philosophical Society, 21, 427-451. |

[16] | Hardy, G.H. (1949) Divergent Series. Oxford University Press, Oxford. (Reprinted in 1973) |

[17] | Blinov, I.N. (1983) Absence of Exact Mean Values for Certain Bounded Functions. Izvestija Akademii Nauk SSSR. Serija Mathematicheskaja (Moscow), 47, 1162-1181. |

[18] | Ditkine, V. and Proudnikov, A. (1979) Calcul Opérationnel. éditions Mir, Moscou. |

[19] | Baumgartel, H. and Wollenberg, M. (1983) Mathematical Scattering Theory. Birkhauser Verlag, Berlin. |

[20] |
Ostrowski, A.M. (1976) On Cauchy-Frullani Integrals. Commentarii Mathematici Helvetici, 51, 57-91. http://dx.doi.org/10.1007/BF02568143 |

[21] |
Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511721434 |

[22] | Hartman, Ph. (1952) On Non-Oscillatory Linear Differential Equations of Second Order. American Journal of Mathematics, 74, 389-400. http://dx.doi.org/10.2307/2372004 |

[23] | Hartman, Ph. (1982) Ordinary Differential Equations. 2nd Edition, Birkhauser, Boston. |

[24] |
Giblin, P.J. (1972) What Is an Asymptote? The Mathematical Gazette, 56, 274-284. http://dx.doi.org/10.2307/3617830 |

Copyright © 2018 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.